Theorem bnj1209

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1209.1	⊢ ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜑 )
	bnj1209.2	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
Assertion	bnj1209	⊢ ( 𝜒 → ∃ 𝑥 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		bnj1209.1	⊢ ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜑 )
2		bnj1209.2	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
3	1	bnj1196	⊢ ( 𝜒 → ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
4	3	ancli	⊢ ( 𝜒 → ( 𝜒 ∧ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
5		19.42v	⊢ ( ∃ 𝑥 ( 𝜒 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ( 𝜒 ∧ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
6	4 5	sylibr	⊢ ( 𝜒 → ∃ 𝑥 ( 𝜒 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
7		3anass	⊢ ( ( 𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑 ) ↔ ( 𝜒 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
8	2 7	bitri	⊢ ( 𝜃 ↔ ( 𝜒 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
9	6 8	bnj1198	⊢ ( 𝜒 → ∃ 𝑥 𝜃 )